Analysis I: Convergence, Elementary functions / Edition 1

Analysis I: Convergence, Elementary functions / Edition 1

by P. Spain, Roger Godement
     
 

ISBN-10: 3540059237

ISBN-13: 9783540059233

Pub. Date: 11/18/2003

Publisher: Springer Berlin Heidelberg

Functions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical

Overview

Functions in R and C, including the theory of Fourier series, Fourier integrals and part of that of holomorphic functions, form the focal topic of these two volumes. Based on a course given by the author to large audiences at Paris VII University for many years, the exposition proceeds somewhat nonlinearly, blending rigorous mathematics skilfully with didactical and historical considerations. It sets out to illustrate the variety of possible approaches to the main results, in order to initiate the reader to methods, the underlying reasoning, and fundamental ideas. It is suitable for both teaching and self-study. In his familiar, personal style, the author emphasizes ideas over calculations and, avoiding the condensed style frequently found in textbooks, explains these ideas without parsimony of words. The French edition in four volumes, published from 1998, has met with resounding success: the first two volumes are now available in English.

Product Details

ISBN-13:
9783540059233
Publisher:
Springer Berlin Heidelberg
Publication date:
11/18/2003
Series:
Universitext Series
Edition description:
Softcover reprint of the original 1st ed. 2004
Pages:
430
Product dimensions:
0.93(w) x 6.14(h) x 9.21(d)

Table of Contents

I Sets and Functions.- §1. Set Theory.- 1 - Membership, equality, empty set.- 2 - The set defined by a relation. Intersections and unions.- 3 - Whole numbers. Infinite sets.- 4 - Ordered pairs, Cartesian products, sets of subsets.- 5 - Functions, maps, correspondences.- 6 - Injections, surjections, bijections.- 7 - Equipotent sets. Countable sets.- 8 - The different types of infinity.- 9 - Ordinals and cardinals.- §2. The logic of logicians.- II - Convergence: Discrete variables.- §1. Convergent sequences and series.- 0 - Introduction: what is a real number?.- 1 - Algebraic operations and the order relation: axioms of—.- 2 - Inequalities and intervals.- 3 - Local or asymptotic properties.- 4 - The concept of limit. Continuity and differentiability.- 5 - Convergent sequences: definition and examples.- 6 - The language of series.- 7 - The marvels of the harmonic series.- 8 - Algebraic operations on limits.- §2. Absolutely convergent series.- 9 - Increasing sequences. Upper bound of a set of real numbers.- 10 - The function log x. Roots of a positive number.- 11 - What is an integral?.- 12 - Series with positive terms.- 13 - Alternating series.- 14 - Classical absolutely convergent series.- 15 - Unconditional convergence: general case.- 16 - Comparison relations. Criteria of Cauchy and d’Alembert.- 17 - Infinite limits.- 18 - Unconditional convergence: associativity.- §3. First concepts of analytic functions.- 19 - The Taylor series.- 20 - The principle of analytic continuation.- 21 - The function cot x and the series—1/n2k.- 22 - Multiplication of series. Composition of analytic functions Formal series.- 23 - The elliptic functions of Weierstrass.- III - Convergence: Continuous variables.- §1. The intermediate value theorem.- 1 - Limit values of a function. Open and closed sets.- 2 - Continuous functions.- 3 - Right and left limits of a monotone function.- 4 - The intermediate value theorem.- §2. Uniform convergence.- 5 - Limits of continuous functions.- 6 - A slip up of Cauchy’s.- 7 - The uniform metric.- 8 - Series of continuous functions. Normal convergence.- §3. Bolzano-Weierstrass and Cauchy’s criterion.- 9 - Nested intervals, Bolzano-Weierstrass, compact sets.- 10 - Cauchy’s general convergence criterion.- 11 - Cauchy’s criterion for series: examples.- 12 - Limits of limits.- 13 - Passing to the limit in a series of functions.- §4. Differentiable functions.- 14 - Derivatives of a function.- 15 - Rules for calculating derivatives.- 16 - The mean value theorem.- 17 - Sequences and series of differentiable functions.- 18 - Extensions to unconditional convergence.- §5. Differentiable functions of several variables.- 19 - Partial derivatives and differentials.- 20 - Differentiability of functions of class C1.- 21 - Differentiation of composite functions.- 22 - Limits of differentiable functions.- 23 - Interchanging the order of differentiation.- 24 - Implicit functions.- Appendix to Chapter III.- 1 - Cartesian spaces and general metric spaces.- 2 - Open and closed sets.- 3 - Limits and Cauchy’s criterion in a metric space; complete spaces.- 4 - Continuous functions.- 5 - Absolutely convergent series in a Banach space.- 6 - Continuous linear maps.- 7 - Compact spaces.- 8 - Topological spaces.- IV Powers, Exponentials, Logarithms, Trigonometric Functions.- §1. Direct construction.- 1 - Rational exponents.- 2 - Definition of real powers.- 3 - The calculus of real exponents.- 4 - Logarithms to base a. Power functions.- 5 - Asymptotic behaviour.- 6 - Characterisations of the exponential, power and logarithmic functions.- 7 - Derivatives of the exponential functions: direct method.- 8 - Derivatives of exponential functions, powers and logarithms.- §2. Series expansions.- 9 - The number e. Napierian logarithms.- 10 - Exponential and logarithmic series: direct method.- 11 - Newton’s binomial series.- 12 - The power series for the logarithm.- 13 - The exponential function as a limit.- 14 - Imaginary exponentials and trigonometric functions.- 15 - Euler’s relation chez Euler.- 16 - Hyperbolic functions.- §3. Infinite products.- 17 - Absolutely convergent infinite products.- 18 - The infinite product for the sine function.- 19 - Expansion of an infinite product in series.- 20 - Strange identities.- §4. The topology of the functions Arg(z) and Log z.

Customer Reviews

Average Review:

Write a Review

and post it to your social network

     

Most Helpful Customer Reviews

See all customer reviews >